where \((z^i)\)  represent (approximate) samples from the
previous filtering marginal
\(\pi _{Z_{k} \vert y_{1:k}}\) . This approximation of the forecast density contrasts with the
regularized maximum likelihood estimation
of the KR rearrangement
\(S\)  in (REF )
from the forecast ensemble,
using
(REF ).
Indeed, it would be interesting to
approximate the rearrangement
\(\smash{\widehat{T}}\) 
in the deterministic map algorithm using the
technique of
[1]}
(or its second-order extension [2]}), after “localizing” the RKHS of transformations by imposing a
sparsity pattern like the one in
(REF ).
Regularization is critical in high dimensions.

